Persistent homology

Summary

Persistent homology studies how the homology of a growing family of spaces changes, and records when each homology class is born and when it dies. The input is a filtration of a space, often built from a finite point cloud at increasing scales. The output is a barcode or persistence diagram, a multiset of intervals that summarises the shape of the data across all scales at once. It is the central tool of topological data analysis.